If a discrete random variable $X$ has the probability distribution $P(X=x) = k \frac{2^{2x+1}}{(2x+1)!}$ for $x = 0, 1, 2, \ldots, \infty$,then $k =$

If $f(x) = \frac{x}{8}$ for $0 < x < 4$ and $f(x) = 0$ otherwise,is the probability density function (p.d.f.) of a continuous random variable $X$,and $F(x)$ is the cumulative distribution function (c.d.f.) associated with $f(x)$,then find $F(0.5)$.

Let $F(x)$ be the cumulative distribution function (c.d.f.) of a continuous random variable $X$. If $F(b) = 0.7$ and $P(X > a) = 0.4$, then the value of $P(a < X < b)$ is ...

$A$ random variable $X$ has the probability distribution
$\begin{array}{|c|c|c|c|c|c|c|}\hline X=x_i & 1 & 2 & 3 & 4 & 5 & 6 \\\hline P(X=x_i) & 0.2 & 0.3 & 0.12 & 0.1 & 0.2 & 0.08 \\\hline \end{array}$
If $A=\{x_i \mid x_i \text{ is a prime number}\}$ and $B=\{x_i \mid x_i < 4\}$ are two events,then $P(A \cup B) = $

Two cards are drawn simultaneously (or successively without replacement) from a well-shuffled pack of $52$ cards. Find the mean,variance,and standard deviation of the number of kings.

$A$ person who tosses an unbiased coin gains two points for turning up a head and loses one point for a tail. If three coins are tossed and the total score $X$ is observed, then the range of $X$ is